Question

In the MIT OCW Linear course: Regarding the recitation video "Computing the Four Fundamental Subspaces" ...

the TA presents a Matrix in LU form and we are asked to find the rank and basis of each of the four fundamental subspaces. I multiplied L by U to get the matrix B, then found the rref of B to be the 3 x 3 identity matrix of rank 3. I computed the bases of the subspaces from this 3 x 3 identity matrix.

What's wrong with this method?

Thanks